On the structure of the group of balanced labelings on graphs

Abstract

Let G = (V, E) be an undirected graph with possible multiple edges and loops (a multigraph). Let A be an Abelian group. In this work we study the following topics:

• 1) A function f:E → A is called balanced if the sum of its values along every closed truncated trail of G is zero. By a truncated trail we mean a trail without the last vertex. The set H(E, A) of all the balanced functions f: E → A is a subgroup of the free Abelian group A ^E of all functions from E to A. We give a full description of the structure of the group H(E, A), and provide an O(¦E¦)-time algorithm to construct a set of the generators of its cyclic direct summands.

• 2) A function g:V → A is called balanceable if there exists some f:E → A such that the sum of all the values of g and f along every closed truncated trail of G is zero. The set B(V, A) of all balanceable functions g:V → A is a subgroup of the free Abelian group A ^V of all the functions from V to A. We give a full description of the structure of the group B(V, A).

• 3) A function h:V ∪ E → A taking values on vertices and edges is called balanced if the sum of its values along every closed truncated trail of G is zero. The set W(V ∪ E, A) of all balanced functions h:V ∪ E → A is a subgroup of the free Abelian group A ^V∪E of all functions from V ∪ E to A. The group H(E, A) is naturally isomorphic to the subgroup of W(V ∪ E, A) consisting of all functions taking every vertex to 0. So we, abusing the notations, treat H(E, A) as that subgroup of W(V ∪ E, A).